Average Error: 13.9 → 9.8
Time: 1.1m
Precision: 64
Internal precision: 128
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^2}\]
⬇
\[\begin{array}{l} \mathbf{if}\;\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \le 1.8969164030706211 \cdot 10^{+155}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\sqrt[3]{-2 \cdot J} \cdot \sqrt[3]{U}\right) \cdot {\left(e^{\frac{1}{3}}\right)}^{\left(\log \frac{1}{2} - \log J\right)}\right)}^3\\ \end{array}\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

  1. Split input into 2 regimes.

  2. if (/ U (* (* 2 J) (cos (/ K 2)))) < 1.8969164030706211e+155

    1. Initial program 10.1

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^2}\]

    if 1.8969164030706211e+155 < (/ U (* (* 2 J) (cos (/ K 2))))

    1. Initial program 60.3

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^2}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt 60.3

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^2}}\right)}^3}\]

    4. Applied add-cube-cbrt 60.3

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^3} \cdot {\left(\sqrt[3]{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^2}}\right)}^3\]

    5. Applied cube-unprod 60.3

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \sqrt[3]{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^2}}\right)}^3}\]

    6. Applied taylor 62.9

      \[\leadsto {\left(e^{\frac{1}{3} \cdot \left(\log -2 + \log J\right)} \cdot e^{\frac{1}{3} \cdot \left(\left(\log U + \log \frac{1}{2}\right) - \log J\right)}\right)}^{3}\]

    7. Taylor expanded around 0 62.9

      \[\leadsto \color{blue}{{\left(e^{\frac{1}{3} \cdot \left(\log -2 + \log J\right)} \cdot e^{\frac{1}{3} \cdot \left(\left(\log U + \log \frac{1}{2}\right) - \log J\right)}\right)}^{3}}\]

    8. Applied simplify 6.1

      \[\leadsto \color{blue}{{\left(\left(\sqrt[3]{-2 \cdot J} \cdot \sqrt[3]{U}\right) \cdot {\left(e^{\frac{1}{3}}\right)}^{\left(\log \frac{1}{2} - \log J\right)}\right)}^3}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 1.1m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(3191742144 694715044 4175323061 3086806877 3769310338 1285397143)'
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  (* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (sqr (/ U (* (* 2 J) (cos (/ K 2)))))))))